Additive category

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In mathematics, specifically in category theory, an additive category is a preadditive category C such that all finite collections of objects A1,...,An of C have a biproduct A1 ⊕ ⋯ ⊕ An in C.

(Recall that a category C is preadditive if all its hom-sets are Abelian groups and composition of morphisms is bilinear; in other words, C is enriched over the monoidal category of Abelian groups. Recall also that a biproduct in a preadditive category is both a finite product and a finite coproduct.)

Warning: The term "additive category" is sometimes applied to any preadditive category, but Wikipedia does not follow this older practice.

Contents

Definition

A category C is additive if

  • it has a zero object
  • every hom-set Hom(A,B) has an addition, endowing it with the structure of an abelian group, and such that composition of morphisms is bilinear
  • all finite biproducts exist.

Note that a category is called preadditive if just the second holds, whereas it is called semiadditive if both the first and the third hold.

Also, since the empty biproduct is a zero object in the category, we may omit the first condition. If we do this, however, we need to presuppose that the category C has zero morphisms, or equivalently that C is enriched over the category of pointed sets.

Examples

The original example of an additive category is the category of abelian groups Ab. The zero object is the trivial group, the addition of morphisms is given point-wise, and biproducts are given by direct sums.

More generally, every module category over a ring R is additive, and so in particular, the category of vector spaces over a field K is additive.

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